超定线性系统最小二乘解的可信验证

The Verification of the Least Square Solution of an Overdetermined Linear System

给定一个系数矩阵为病态矩阵的超定线性系统,本文利用区间算法和增广矩阵技术,设计算法输出给定超定线性系统的微小摄动区间系统,算法保证在输出的区间系统中存在系数阵精确秩亏的超定线性系统,算法输出该系数阵精确秩亏的超定线性系统的最小二乘解的高精度近似解和其相应的可信误差界。

Given an overdetermined linear system with ill-conditioned coefficient matrix, this paper mainly uses interval algorithm and augmented matrix technology to design an algorithm to output a minimally perturbed interval system for the given overdetermined linear system. The algorithm guarantees that there is an overdetermined linear system with a coefficient matrix of exactly rank deficiency in the output interval system. The algorithm also outputs the high-precision approximate solution of the least square solution of the overdetermined linear system and corresponding verified error bound.